Answer
$V(x)= 4x^{3}-64x^{2}+240x$
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Work Step by Step
From the figure, we can see that the box will have a length $20 - 2x$ in, a width $12 - 2x$, and a height $x$. The volume of a rectangular box is found by multiplying its length by its width by its height. Therefore, the volume $V$ of the box as a function of $x$ will be:
$V(x) = (20 - 2x)(12 - 2x)x$
Using the FOIL method, we get:
$V(x) = (20 - 2x)(12 - 2x)x$
$= x(240 - 40x-24x+4x^{2})$
$= x(4x^{2}-64x+240)$
$= 4x^{3}-64x^{2}+240x$
$x$ can not equal anything that would cause the box width to equal zero or negative. Evaluate the width equation to determine the maximum length of $x$.
$0=12-2x$
$12=2x$
$x=6$
Therefore $x$ must be less than $6$ and greater than $0$.
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